| L(s) = 1 | + (−0.138 − 1.40i)2-s + (−1.96 + 0.389i)4-s + (−0.608 + 1.67i)5-s + (0.408 − 2.31i)7-s + (0.820 + 2.70i)8-s + (2.43 + 0.624i)10-s + (0.860 + 2.36i)11-s + (0.359 + 0.428i)13-s + (−3.31 − 0.253i)14-s + (3.69 − 1.52i)16-s + (−1.49 − 2.59i)17-s + (6.64 + 3.83i)19-s + (0.541 − 3.51i)20-s + (3.20 − 1.53i)22-s + (1.12 + 6.39i)23-s + ⋯ |
| L(s) = 1 | + (−0.0979 − 0.995i)2-s + (−0.980 + 0.194i)4-s + (−0.272 + 0.747i)5-s + (0.154 − 0.874i)7-s + (0.290 + 0.956i)8-s + (0.770 + 0.197i)10-s + (0.259 + 0.712i)11-s + (0.0997 + 0.118i)13-s + (−0.885 − 0.0678i)14-s + (0.923 − 0.382i)16-s + (−0.363 − 0.629i)17-s + (1.52 + 0.880i)19-s + (0.121 − 0.786i)20-s + (0.683 − 0.327i)22-s + (0.235 + 1.33i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20856 - 0.377199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20856 - 0.377199i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.138 + 1.40i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.608 - 1.67i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.408 + 2.31i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.860 - 2.36i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.359 - 0.428i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.49 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.64 - 3.83i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.12 - 6.39i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.22 + 6.22i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.643 + 3.65i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.07 + 2.35i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.67 + 4.76i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.928 - 2.55i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0919 - 0.521i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 6.39iT - 53T^{2} \) |
| 59 | \( 1 + (2.87 - 7.88i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.716 - 0.126i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.68 - 4.39i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (7.38 + 12.7i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.98 - 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.94 + 7.50i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (4.81 - 5.74i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.16 + 2.02i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.8 + 5.05i)T + (74.3 - 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47013687692734842106290997352, −9.814472231990977725950530162764, −9.085130081200160389848980983863, −7.54751324975566788215779105402, −7.43614394580341594356672671950, −5.82718989121485559202282878735, −4.55108713633102700257884658129, −3.73930167360538921596911235554, −2.69305073549330007810321872176, −1.18142630384239624558832240366,
0.941219951037905073719196057740, 3.07093005484320774133020310469, 4.52041024028207014204494971136, 5.20482215359926891656842612466, 6.15802951256226958012786129327, 7.04607542533853859571613122742, 8.302473299307391452576432413284, 8.648327614755971723807254732415, 9.354684580930100507017531347495, 10.50815580961955893562271342796
